Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

‘there is no ontology here’ 389the two real algebraic solutions√ √2/2, 2/2 ∈ R and√ √− 2/2, − 2/2 ∈ Rand another point combining these two¹⁸√2/2, −√2/2 ∈ R and −√2/2,√2/2 ∈ RIt has one point for each pair of conjugate complex algebraic solutions such as2, ± √ −3 ∈ CIt has no points for real or complex transcendental solutions. This elegantalgebraic definition gives our space points for precisely those solutions toEquation (14.3) given by roots of integer polynomials (possibly modulo someprime number) and it distinguishes only those points given by roots of distinctpolynomials.Intuitively a closed set should be the set of all points where some function is 0,or where some list of functions are all 0. Formally, an affine scheme has a closedset for each ideal of coordinate functions on it, and in the case of arithmeticschemes each ideal is defined by a finite list of polynomial equations:P 1 (X, Y) = 0 ... P k (X, Y) = 0Then we name the space after its coordinate ring, calling itSpec(Z[X, Y]/(X 2 + Y 2 − 1))or the spectrum of the ring Z[X, Y]/(X 2 + Y 2 − 1).Every commutative ring R has a spectrum Spec(R). The coordinate functionring on Spec(R) is just the ring R. So again the ‘functions’ are generally notfunctions in the set-theoretic sense. They are any elements of any ring. Thepoints are the prime ideals of R. The spectrum has a topology where closed setscorrespond to ideals. The spectra of rings are the affine schemes. Notably, theaffine n-space A n Zis the spectrum of the integer polynomial ring in n variablesZ[X 1 , ... X n ] subject to no equation or, if you prefer, the trivially true equation1 = 1.A n Z = Spec(Z[X 1, ... X n ])A scheme is patched together from affine schemes. More fully, a scheme is atopological space X together with a sheaf of rings O X called the structure sheaf¹⁸ Notice the first two solutions also satisfy 2XY = 1. The second two satisfy 2XY =−1.